, 92:29 | Cite as

Synchronisation of fractional-order complex systems and its application

  • Milad Mohadeszadeh
  • Ali Karimpour
  • Naser ParizEmail author


In this paper, a passive control scheme based on the fractional-order calculus is proposed. We study the modified complex projective synchronisation between two identical fractional-order complex chaotic systems, and its application in the secure communication. The fractional-order complex chaotic Lorenz system is employed to encrypt the emitted signal. In the transmitter module, the information signal is modulated into one parameter of the Lorenz system. It is assumed that the same parameter is unknown in the receiver module. In order to synchronise two systems with different initial conditions, the controllers and an appropriate parameter update rule are designed. Theoretical analysis and numerical simulations show that this method is feasible and robust to some extent in the presence of channel noise.


Fractional-order chaotic system complex projective synchronisation parameter modulation chaotic secure communication 


02.30.Yy 05.45.–a 05.45.Gg 05.45.Vx 



N Pariz, the corresponding author, was supported by a grant from Ferdowsi University of Mashhad (No. 44878).


  1. 1.
    T Skovranek, I Pudlubny and I Petras, Econ. Model. 2, 1322 (2012)CrossRefGoogle Scholar
  2. 2.
    N Noghredani, A Riahi, N Pariz and A Karimpour, Pramana – J. Phys. 90: 26 (2018)ADSCrossRefGoogle Scholar
  3. 3.
    X J Wu, H Wang and H Lu, Nonlinear Anal. Real World Appl. 13, 1441 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S Liu and F F Zhang, Nonlinear Dyn. 76, 1087 (2014)CrossRefGoogle Scholar
  5. 5.
    G M Mahmoud, E E Mahmoud and A A Arafa, Phys. Scr. 87, 1 (2013)CrossRefGoogle Scholar
  6. 6.
    X Wu, C Zhu and H Kan, Appl. Math. Comput. 252, 201 (2015)MathSciNetGoogle Scholar
  7. 7.
    C Luo and X Wang, Nonlinear Dyn. 71, 241 (2013)CrossRefGoogle Scholar
  8. 8.
    C Luo and X Wang, Int. J. Mod. Phys. C 24, 1 (2013)CrossRefGoogle Scholar
  9. 9.
    X J Liu, L Hong and L X Yang, Nonlinear Dyn. 75, 589 (2014)CrossRefGoogle Scholar
  10. 10.
    C Jiang, S Liu and C Luo, Hindawi 2014, 326354 (2014)Google Scholar
  11. 11.
    L M Pecora and T L Carroll, Phys. Rev. Lett. 64, 821 (1990)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    K Rabah, S Ladaci and M Lashab, Pramana – J. Phys. 89, 46 (2017)ADSCrossRefGoogle Scholar
  13. 13.
    H Delavari and M Mohadeszadeh, ASME J. Comput. Nonlinear Dyn. 11, 041023-1 (2016).CrossRefGoogle Scholar
  14. 14.
    K Vishal, S K Agrawal and S Das, Pramana – J. Phys. 86, 59 (2016)Google Scholar
  15. 15.
    G M Mahmoud and E E Mahmoud, Nonlinear Dyn. 73, 2231 (2013)CrossRefGoogle Scholar
  16. 16.
    M Lakshmanan and K Murali, Chaos in nonlinear oscillators, controlling and synchronization (World Scientific, Singapore, 1996)CrossRefGoogle Scholar
  17. 17.
    B Blasius, A Huppert and L Stone, Nature 399, 354 (1999)ADSCrossRefGoogle Scholar
  18. 18.
    C J Cheng, Appl. Math. Comput. 219, 2698 (2012)MathSciNetGoogle Scholar
  19. 19.
    C Li and Y Tong, Pramana – J. Phys. 80, 583 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    A Nourian and S Balochian, Pramana – J. Phys. 86, 1401 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    R Shahnazi, N Pariz and A Vahidian Kamyad, Asian J. Control 13, 456 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M Mohadeszadeh and H Delavari, Int. J. Dynam. Control 5, 135 (2015)CrossRefGoogle Scholar
  23. 23.
    M P Aghababa, IET Sci. Meas. Technol. 9, 122 (2015)Google Scholar
  24. 24.
    F Wang and C Liu, Physica D 225, 55 (2007)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    F Q Wang and C X Liu, Phys. D: Nonlinear Phenom. 225, 55 (2007)ADSCrossRefGoogle Scholar
  26. 26.
    T Yang, Int. J. Comput. Cogn. 2, 81 (2004)Google Scholar
  27. 27.
    A Kiani-B, K Fallahi, N Pariz and H Leung, Commun. Nonlinear Sci. Numer. Simul. 14, 863 (2009)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    B Naderi and H Kheiri, Int. J. Light Electron. Opt. 127, 2407 (2016)CrossRefGoogle Scholar
  29. 29.
    A H Mazinan, M F Kazemi and H Shirzad, Trans. Inst. Meas. Control 36, 164 (2014)CrossRefGoogle Scholar
  30. 30.
    H Dedieu, M P Kennedy and M Hasler, IEEE Trans. Circuits Syst. II 40, 634 (1993)CrossRefGoogle Scholar
  31. 31.
    J S Lin, C F Huang and T L Liao, Digit. Signal Process 20, 229 (2010)CrossRefGoogle Scholar
  32. 32.
    I Podlubny, Fractional differential equations (Academic Press, New York, 1999)zbMATHGoogle Scholar
  33. 33.
    A A Kilbas, H M Srivastava and J J Trujillo, Theory and applications of fractional differential equations (Elsevier Science Inc., New York, 2006)zbMATHGoogle Scholar
  34. 34.
    K Diethelm, N J Ford and A D Freed, Numer. Algorithms 36, 31 (2004)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    C I Byrnes, A Isidori and J C Willem, IEEE Trans. Autom. Control 36, 1228 (1991)CrossRefGoogle Scholar
  36. 36.
    D J Hill and P J Moylan, IEEE Trans. Autom. Control 21, 708 (1976)CrossRefGoogle Scholar
  37. 37.
    A C Norelys, D M A Manuel and A G Javier, Commun. Nonlinear Sci. Numer. Simul. 19, 2951 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  • Milad Mohadeszadeh
    • 1
  • Ali Karimpour
    • 1
  • Naser Pariz
    • 1
    Email author
  1. 1.Department of Electrical Engineering, Faculty of EngineeringFerdowsi University of Mashhad (FUM) CampusMashhadIran

Personalised recommendations